The two phases of CsGeBr₃ (cubic and tetragonal) were typically chosen as the highest temperature phases of the cubic structure to carry out the numerical computations for high-temperature device applications.

The perovskite structure is composed of corner sharing tetragonal and A-site cations in the middle of four neighbouring tetrahedra, as shown in Figure 2. The versatile perovskite structure can incorporate a large array of elements, making it possible to align its physical properties through substitutional alloying. In this perovskite, the Murnaghan Equation of State (EoS)was used to obtain the equilibrium volume, the bulk modulus and its derivatives. From thermodynamics definition, an EoS relates the volume, temperature and pressure of a material in a thermodynamic equilibrium [16] [17] . The relationship between the pressure

and volume of a material with fixed number of particles is given by Equation (1) [17] .

P = 1.5 B o [ ( v o v ) 7 3 − ( v o v ) 5 3 ] [ 1 + 0.75 ( B ′ o − 4 ) ( { v o v } 2 3 − 1 ) ] (1)

To find the energy with respect to the volume, Equation (4.1) is integrated leading to Equation (2).

E ( V ) = E o + 9 V o B o 16 { [ ( V o V ) 2 3 − 1 ] 3 B ′ o + [ ( V o V ) 2 3 − 1 ] 2 [ 6 − 4 ( V o V ) 2 3 ] } (2)

In Quantum Espresso, the energy/volume data computed were fitted using the Murnaghan equation. This fitting was done after the lattice constant convergence was carried out from which the energies of different volumes were obtained as in Figure 3; This figure shows that the minimum energy is −2930.41 eV for CsGeBr₃ cubic structure showing that it is the most stable of the perovskites under study.

The volume vs energy curves are shown in Figure 3. Y1, Y2 and Y3 helps us to derive an equillibrium latice constant and the bulk modulus for the cubic and tetragonal phases of CsGeBr₃ and CsSiBr₃. The volume here is a representative of of the unit cell area. The minimum free energy for the cubic phase of CsSiBr and CsGeBr₃ is −2950 Ry −1900 Ry and −3200 Ry for the tetragonal phase of CsSiBr₃. This infers that the cubic phase is more stable than the tetragonal and monoclinic phases. This values are significant in obtaining the corresponding bulk modulus. This work used the Murnaghan (EoS) to get the equilibrium bulk modulus and its pressure derivative. It gauges a material’s resistance to compression. According to [18] , the volume and bulk modulus of materials are inversely connected. Equation (1) provides the bulk modulus (3).

B = − V ∂ P ∂ V (3)

where B is the bulk modulus, the volume and P is the pressure. The pressure is given by Equation (4) below;

P = − ∂ E ∂ V (4)

Equation (4) then reduces (3) into Equation (5)

B = V − ∂ 2 E ∂ V 2 (5)

The pressure derivative of the bulk modulus is given by

B ′ = ∂ B ∂ P = 1 B ( V ∂ ∂ V ( V − ∂ 2 E ∂ V 2 ) ) (6)

The relationship between pressure and volume for CsGeBr₃ is shown in the graph below. The position where the two lines meet indicates the minimum volume of convergence. CsGeBr₃ gives 1190 (a.u)³ and CsSiBr₃ gives 1115 (a.u)³ in cubic phases.

The extent of similarity is clear when pressure-volume curves in Figure 4 and Figure 5 where “pressure” is the scalar hydrostatic pressure on the periodic system. As this pressure P = −dE/dV, the optimal geometries dE/dV = 0 are now those intersecting the P = 0 line. The slope of one P-V curve may be used to estimate the crossing point of another hence the similarity in the P-V curvature.

Elastic properties are significant in determining the mechanical stability of compounds in solid states. The properties studied in this work are the elastic constants, the bulk modulus, Young and shear moduli and the Poisson’s ratio. The elastic properties were calculated using the thermo_pw within Q.E. The

elastic constants are denoted by C_ij and varies from one crystal system to the other. From the Born stability criteria [19] , the elastic constants need to meet certain conditions and be positive for the materials to be considered mechanically stable. In the cubic phase, the necessary conditions are given by Equations (7) [19] [20] [21] [22] .

C 11 − C 12 > 0 , C 11 + 2 C 12 > 0 , C 44 > 0 (7)

A tetragonal material is regarded mechanically stable if Equations (8) are met [19] [23] ;

C 11 > C 12 ∨ , 2 C 13 2 < C 33 ( C 11 + C 12 ) , C 44 > 0 (8)

After the calculation of the elastic constants, the moduli are calculated using the Reuss and Voigt theory and their average obtained from the Hill averaging scheme [10] . From the C_ij’s, and using the Voigt theory, the bulk B_V and shear G_V moduli are given by Equations (9) and (10) respectively [23] .

B V = 1 9 [ ( C 11 + C 22 + C 33 ) + 2 ( C 12 + C 23 + C 31 ) ] (9)

G V = 1 15 [ ( C 11 + C 22 + C 33 ) − ( C 12 + C 23 + C 31 ) + 3 ( C 44 + C 55 + C 66 ) ] (10)

From the Reuss theory, the bulk B_R and shear G_R moduli are obtained from the elastic constants using Equations (11) and (12) respectively.

1 B R = ( S 11 + S 22 + S 33 ) + 2 ( S 12 + S 23 + S 31 ) (11)

15 G R = 4 ( S 11 + S 22 + S 33 ) − 4 ( S 12 + S 23 + S 31 ) + 3 ( S 44 + S 55 + S 66 ) (12)

S_ij’s are the elastic compliances and are the inverse of the elastic constant’s matrix. From the Hill averaging scheme, the bulk B_H and shear G_H moduli are given by the averages of the two theories, that is Equation (13) and (14) respectively.

B H = B V + B R 2 (13)

G H = G V + G R 2 (14)

From the Pugh method [23] brittleness or ductility of a material can be obtained by checking the relation of B G to the critical value of 1.75. If the ratio B G is more than the value, the material is considered ductile and vice versa.

After calculating the moduli and the elastic constants, the Poison’s ratio and the young’s modulus are obtained from the shear and bulk moduli using Equations (15) and (16) respectively [19] [20] [21] [22] [23] .

v = 3 B H − 2 G H 2 ( 3 B H + G H ) (15)

E H = 9 B H G H 3 B H + G H (16)

The cubic and tetragonal perovskites satisfy the Born stability criterion hence they are mechanically stable [20] . The values of elastic constants for the cubic and tetragonal phases are given in Table 1.

The shear and moduli in the Voigt and Reuss approximations and their averages in GPa are shown in the Table 2. The values of B_V, G_V, B_R, G_R, B_H, G_H are obtained from Equations (9)-(14) respectively. The cubic structures had the lowest bulk moduli with the germanium compounds having low moduli than their silicon counterparts. We also noted that the monoclinic structures had inconsistent figures of both the moduli and elastic constants, showing that they are highly elastically unstable.

The Pugh criterion [5] , which is employed in determining the ductile or brittle nature of materials was employed, where a material is ductile if B G > 1.75 and brittle otherwise. From Table 3, the B G ratios obtained implies that the germanium-containing perovskites are ductile while the materials containing silicon are brittle because their B G ratios are less than the critical value of 1.75. The

data obtained from the Poisson ratio also agree with these findings. Using the critical value of the Poisson’s ratio, that is, 0.26, the materials containing germanium in the B site have ν > 0.26 showing that they are ductile in nature. The perovskite materials with silicon in the B site have ν < 0.26 indicating that they are brittle with the monoclinic structure having the lowest value of 0.074 < 0.26 showing that it is highly brittle; even more than the cubic and tetragonal materials. From the calculations, the Pugh’s criterion and the Poisson ratio are consistent in identifying the ductility of materials. This implies that germanium perovskite structures would crack easily especially during materials engineering for applications.

Figure 6(a) depicts the band structure of CsXBr₃ in the BQ at ambient pressure with an FM spin configura tion in the high symmetry direction. X could be either Ge or Si. A coloured line denotes the Fermi level (EF), which is set to zero. The C-2s and Cr-3d states make up the majority of the bands in the energy range from −15 eV to −12 eV, suggesting that the Cr-3d electrons are itinerant. At practically every level, overlaps exist on top of the up- and down-spin states.